BLUMAN A. G. Probability Demystified.pdf

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PROBABILITY DEMYSTIFIED

PROBABILITY DEMYSTIFIED

ALLAN G BLUMAN

McGRAW-HILL

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To all of my teachers, whose examples instilled in me my love ofmathematics and teaching.

CHAPTER 8 Other Probability Distributions 131

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‘‘The probable is what usually happens.’’ — Aristotle

Probability can be called the mathematics of chance The theory of ity is unusual in the sense that we cannot predict with certainty the individualoutcome of a chance process such as flipping a coin or rolling a die (singularfor dice), but we can assign a number that corresponds to the probability ofgetting a particular outcome For example, the probability of getting a headwhen a coin is tossed is 1/2 and the probability of getting a two when a singlefair die is rolled is 1/6

probabil-We can also predict with a certain amount of accuracy that when a coin istossed a large number of times, the ratio of the number of heads to the totalnumber of times the coin is tossed will be close to 1/2

Probability theory is, of course, used in gambling Actually, cians began studying probability as a means to answer questions aboutgambling games Besides gambling, probability theory is used in many otherareas such as insurance, investing, weather forecasting, genetics, and medicine,and in everyday life

mathemati-What is this book about?

First let me tell you what this book is not about:

This book is not a rigorous theoretical deductive mathematicalapproach to the concepts of probability

This book is not a book on how to gamble

And most important

ix

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This book is not a book on how to win at gambling!

This book presents the basic concepts of probability in a simple,straightforward, easy-to-understand way It does require, however, aknowledge of arithmetic (fractions, decimals, and percents) and a knowledge

of basic algebra (formulas, exponents, order of operations, etc.) If you need

a review of these concepts, you can consult another of my books in thisseries entitled Pre-Algebra Demystified

This book can be used to gain a knowledge of the basic concepts ofprobability theory, either as a self-study guide or as a supplementarytextbook for those who are taking a course in probability or a course instatistics that has a section on probability

The basic concepts of probability are explained in the first two chapters.Then the addition and multiplication rules are explained Followingthat, the concepts of odds and expectation are explained The countingrules are explained in Chapter 6, and they are needed for the binomial andother probability distributions found in Chapters 7 and 8 The relationshipbetween probability and the normal distribution is presented in Chapter 9.Finally, a recent development, the Monte Carlo method of simulation, isexplained in Chapter 10 Chapter 11 explains how probability can be used ingame theory and Chapter 12 explains how probability is used in actuarialscience Special material on Bayes’ Theorem is presented in the Appendixbecause this concept is somewhat more difficult than the other conceptspresented in this book

In addition to addressing the concepts of probability, each chapter endswith what is called a ‘‘Probability Sidelight.’’ These sections cover some ofthe historical aspects of the development of probability theory or somecommentary on how probability theory is used in gambling and everyday life

I have spent my entire career teaching mathematics at a level that moststudents can understand and appreciate I have written this book with thesame objective in mind Mathematical precision, in some cases, has beensacrificed in the interest of presenting probability theory in a simplified way.Good luck!

Allan G Bluman

PREFACE

x

I would like to thank my wife, Betty Claire, for helping me with the tion of this book and my editor, Judy Bass, for her assistance in its pub-lication I would also like to thank Carrie Green for her error checkingand helpful suggestions

prepara-xi

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CHAPTER 1

Basic Concepts

Introduction

Probability can be defined as the mathematics of chance Most people are

familiar with some aspects of probability by observing or playing gambling

games such as lotteries, slot machines, black jack, or roulette However,

probability theory is used in many other areas such as business, insurance,

weather forecasting, and in everyday life

In this chapter, you will learn about the basic concepts of probability using

various devices such as coins, cards, and dice These devices are not used as

examples in order to make you an astute gambler, but they are used because

they will help you understand the concepts of probability

1

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Probability Experiments

Chance processes, such as flipping a coin, rolling a die (singular for dice), ordrawing a card at random from a well-shuffled deck are called probabilityexperiments A probability experiment is a chance process that leads to well-defined outcomes or results For example, tossing a coin can be considered

a probability experiment since there are two well-defined outcomes—headsand tails

An outcome of a probability experiment is the result of a single trial of

a probability experiment A trial means flipping a coin once, or drawing asingle card from a deck A trial could also mean rolling two dice at once,tossing three coins at once, or drawing five cards from a deck at once

A single trial of a probability experiment means to perform the experimentone time

The set of all outcomes of a probability experiment is called a samplespace Some sample spaces for various probability experiments are shownhere

Experiment Sample Space Toss one coin H, T*

Roll a die 1, 2, 3, 4, 5, 6 Toss two coins HH, HT, TH, TT

*H = heads; T = tails.

Notice that when two coins are tossed, there are four outcomes, not three.Consider tossing a nickel and a dime at the same time Both coins could fallheads up Both coins could fall tails up The nickel could fall heads up andthe dime could fall tails up, or the nickel could fall tails up and the dimecould fall heads up The situation is the same even if the coins areindistinguishable

It should be mentioned that each outcome of a probability experimentoccurs at random This means you cannot predict with certainty whichoutcome will occur when the experiment is conducted Also, each outcome

of the experiment is equally likely unless otherwise stated That means thateach outcome has the same probability of occurring

When finding probabilities, it is often necessary to consider severaloutcomes of the experiment For example, when a single die is rolled, youmay want to consider obtaining an even number; that is, a two, four, or six.This is called an event An event then usually consists of one or more

CHAPTER 1 Basic Concepts

2

outcomes of the sample space (Note: It is sometimes necessary to consider

an event which has no outcomes This will be explained later.)

An event with one outcome is called a simple event For example, a die is

rolled and the event of getting a four is a simple event since there is only one

way to get a four When an event consists of two or more outcomes, it is

called a compound event For example, if a die is rolled and the event is getting

an odd number, the event is a compound event since there are three ways to

get an odd number, namely, 1, 3, or 5

Simple and compound events should not be confused with the number of

times the experiment is repeated For example, if two coins are tossed, the

event of getting two heads is a simple event since there is only one way to get

two heads, whereas the event of getting a head and a tail in either order is

a compound event since it consists of two outcomes, namely head, tail and

tail, head

EXAMPLE: A single die is rolled List the outcomes in each event:

a Getting an odd number

b Getting a number greater than four

c Getting less than one

SOLUTION:

a The event contains the outcomes 1, 3, and 5

b The event contains the outcomes 5 and 6

c When you roll a die, you cannot get a number less than one; hence,

the event contains no outcomes

Classical Probability

Sample spaces are used in classical probability to determine the numerical

probability that an event will occur The formula for determining the

probability of an event E is

PðEÞ ¼number of outcomes contained in the event E

total number of outcomes in the sample space

CHAPTER 1 Basic Concepts 3

EXAMPLE: Two coins are tossed; find the probability that both coins landheads up.

SOLUTION:

The sample space for tossing two coins is HH, HT, TH, and TT Since thereare 4 events in the sample space, and only one way to get two heads (HH),the answer is

PðHHÞ ¼1

4

EXAMPLE: A die is tossed; find the probability of each event:

a Getting a two

b Getting an even number

c Getting a number less than 5

b Pðblack or pinkÞ ¼3 þ 4

20 ¼

720

c P(not yellow) = P(red or black or pink) ¼8 þ 3 þ 4

20 ¼

15

20¼

34

d P(orange)=0

20¼0, since there are no orange jellybeans.

Probabilities can be expressed as reduced fractions, decimals, or percents

For example, if a coin is tossed, the probability of getting heads up is 1

2 or0.5 or 50% (Note: Some mathematicians feel that probabilities should

be expressed only as fractions or decimals However, probabilities are often

given as percents in everyday life For example, one often hears, ‘‘There is a

50% chance that it will rain tomorrow.’’)

Probability problems use a certain language For example, suppose a die

is tossed An event that is specified as ‘‘getting at least a 3’’ means getting a

3, 4, 5, or 6 An event that is specified as ‘‘getting at most a 3’’ means getting

a 1, 2, or 3

Probability Rules

There are certain rules that apply to classical probability theory They are

presented next

Rule 1: The probability of any event will always be a number from zero to one

This can be denoted mathematically as 0
P(E)
1 What this means is that

all answers to probability problems will be numbers ranging from zero to

one Probabilities cannot be negative nor can they be greater than one

Also, when the probability of an event is close to zero, the occurrence of

the event is relatively unlikely For example, if the chances that you will win a

certain lottery are 0.00l or one in one thousand, you probably won’t win,

unless of course, you are very ‘‘lucky.’’ When the probability of an event is

0.5 or 12, there is a 50–50 chance that the event will happen—the same

CHAPTER 1 Basic Concepts 5

probability of the two outcomes when flipping a coin When the probability

of an event is close to one, the event is almost sure to occur For example,

if the chance of it snowing tomorrow is 90%, more than likely, you’ll seesome snow See Figure 1-1

Rule 2: When an event cannot occur, the probability will be zero

EXAMPLE: A die is rolled; find the probability of getting a 7

SOLUTION:

Since the sample space is 1, 2, 3, 4, 5, and 6, and there is no way to get a 7,P(7) ¼ 0 The event in this case has no outcomes when the sample space isconsidered

Rule 3: When an event is certain to occur, the probability is 1

EXAMPLE: A die is rolled; find the probability of getting a number lessthan 7

Rule 5: The probability that an event will not occur is equal to 1 minus theprobability that the event will occur.

For example, when a die is rolled, the sample space is 1, 2, 3, 4, 5, 6.Now consider the event E of getting a number less than 3 This eventconsists of the outcomes 1 and 2 The probability of event E isPðEÞ ¼26¼13 The outcomes in which E will not occur are 3, 4, 5, and 6, sothe probability that event E will not occur is 4

6¼23 The answer can also

be found by substracting from 1, the probability that event E will occur.That is, 1 1

3¼23

If an event E consists of certain outcomes, then event E (E bar) is called thecomplement of event E and consists of the outcomes in the sample spacewhich are not outcomes of event E In the previous situation, the outcomes in

Eare 1 and 2 Therefore, the outcomes in E are 3, 4, 5, and 6 Now rule fivecan be stated mathematically as

PðEÞ ¼1  PðEÞ:

EXAMPLE: If the chance of rain is 0.60 (60%), find the probability that itwon’t rain

SOLUTION:

Since P(E) = 0.60 and PðEÞ ¼ 1  PðEÞ, the probability that it won’t rain is

1  0.60 = 0.40 or 40% Hence the probability that it won’t rain is 40%

PRACTICE

1 A box contains a $1 bill, a $2 bill, a $5 bill, a $10 bill, and a $20 bill

A person selects a bill at random Find each probability:

a The bill selected is a $10 bill

b The denomination of the bill selected is more than $2

c The bill selected is a $50 bill

d The bill selected is of an odd denomination

e The denomination of the bill is divisible by 5

CHAPTER 1 Basic Concepts 7

2 A single die is rolled Find each probability:

a The number shown on the face is a 2

b The number shown on the face is greater than 2

c The number shown on the face is less than 1

d The number shown on the face is odd

3 A spinner for a child’s game has the numbers 1 through 9 evenlyspaced If a child spins, find each probability:

a The number is divisible by 3

b The number is greater than 7

c The number is an even number

4 Two coins are tossed Find each probability:

a Getting two tails

b Getting at least one head

c Getting two heads

5 The cards A˘, 2^, 3¨, 4˘, 5¯, and 6¨ are shuffled and dealt face down

on a table (Hearts and diamonds are red, and clubs and spades areblack.) If a person selects one card at random, find the probability thatthe card is

8 On a roulette wheel there are 38 sectors Of these sectors, 18 are red,

18 are black, and 2 are green When the wheel is spun, find theprobability that the ball will land on

a Red

b Green

9 A person has a penny, a nickel, a dime, a quarter, and a half-dollar

in his pocket If a coin is selected at random, find the probability thatthe coin is

a A quarter

b A coin whose amount is greater than five cents

c A coin whose denomination ends in a zero

10 Six women and three men are employed in a real estate office If a person

is selected at random to get lunch for the group, find the probabilitythat the person is a man

ANSWERS

1 The sample space is $1, $2, $5, $10, $20

a P($10) =1

5.

b P(bill greater than $2) =3

5, since $5, $10, and $20 are greaterthan $2

6, since there is only one 2 in the sample space.

b P(number greater than 2) =4

c P(number less than 1) =0

6¼0, since there are no numbers in thesample space less than 1

d P(number is an odd number) =3

b P(number greater than 7) =2

9, since 8 and 9 are greater than 7.

c P(even number) =4

9, since 2, 4, 6, and 8 are even numbers.

4 The sample space is HH, HT, TH, TT

a P(TT) =1

4, since there is only one way to get two tails.

b P(at least one head) =3

4, since there are three ways (HT, TH, HH)

to get at least one head

c P(HH) =1

4, since there is only one way to get two heads.

5 The sample space is A˘, 2^, 3¨, 4˘, 5¯, 6¨

3, since there are two clubs.

6 The sample space is red, blue, green, and white

6¼0, since there is no pink ball.

7 The sample space consists of the letters in ‘‘computer.’’

8, since o, u, and e are the vowels in the word.

CHAPTER 1 Basic Concepts

10

8 There are 38 outcomes:

Probabilities can be computed for situations that do not use sample spaces

In such cases, frequency distributions are used and the probability is called

empirical probability For example, suppose a class of students consists of

4 freshmen, 8 sophomores, 6 juniors, and 7 seniors The information can be

summarized in a frequency distribution as follows:

Rank Frequency Freshmen 4 Sophomores 8 Juniors 6 Seniors 7 TOTAL 25

From a frequency distribution, probabilities can be computed using the

following formula

CHAPTER 1 Basic Concepts 11

PðEÞ ¼ frequency of E

sum of the frequencies

Empirical probability is sometimes called relative frequency probability

EXAMPLE: Using the frequency distribution shown previously, find theprobability of selecting a junior student at random

Several things should be explained here First of all, the 1004 peopleconstituted a sample selected from a larger group called the population.Second, the exact probability for the population can never be known unlessevery single member of the group is surveyed This does not happen in thesekinds of surveys since the population is usually very large Hence, the 17% isonly an estimate of the probability However, if the sample is representative

of the population, the estimate will usually be fairly close to the exactprobability Statisticians have a way of computing the accuracy (called themargin of error) for these situations For the present, we shall justconcentrate on the probability

Also, by a representative sample, we mean the subjects of the sample havesimilar characteristics as those in the population There are statisticalmethods to help the statisticians obtain a representative sample Thesemethods are called sampling methods and can be found in many statisticsbooks

EXAMPLE: The same study found 7% considered George Washington to bethe greatest President If a person is selected at random, find the probabilitythat he or she considers George Washington to be the greatest President

CHAPTER 1 Basic Concepts

12

The probability is 7%

EXAMPLE: In a sample of 642 people over 25 years of age, 160 had a

bachelor’s degree If a person over 25 years of age is selected, find the

probability that the person has a bachelor’s degree

SOLUTION:

In this case,

Pðbachelor’s degreeÞ ¼160

642¼0:249 or about 25%:

EXAMPLE: In the sample study of 642 people, it was found that 514 people

have a high school diploma If a person is selected at random, find the

probability that the person does not have a high school diploma

SOLUTION:

The probability that a person has a high school diploma is

P(high school diploma) ¼514

642¼0:80 or 80%:

Hence, the probability that a person does not have a high school diploma is

P ðno high school diplomaÞ ¼ 1  Pðhigh school diplomaÞ

¼1  0:80 ¼ 0:20 or 20%:

Alternate Solution:

If 514 people have a high school diploma, then 642  514 ¼ 128 do not have a

high school diploma Hence

Pðno high school diplomaÞ ¼128

642¼0:199 or 20% rounded:

Consider another aspect of probability Suppose a baseball player has a

batting average of 0.250 What is the probability that he will get a hit the next

time he gets to bat? Although we cannot be sure of the exact probability, we

can use 0.250 as an estimate Since 0:250 ¼14, we can say that there is a one

in four chance that he will get a hit the next time he bats

CHAPTER 1 Basic Concepts 13

1 A recent survey found that the ages of workers in a factory is uted as follows:

distrib-Age Number 20–29 18 30–39 27 40–49 36 50–59 16

60 or older 3 Total 100

If a person is selected at random, find the probability that the person is

a 40 or older

b Under 40 years old

c Between 30 and 39 years old

d Under 60 but over 39 years old

2 In a sample of 50 people, 19 had type O blood, 22 had type A blood,

7 had type B blood, and 2 had type AB blood If a person isselected at random, find the probability that the person

a Has type A blood

b Has type B or type AB blood

c Does not have type O blood

d Has neither type A nor type O blood

3 In a recent survey of 356 children aged 19–24 months, it was foundthat 89 ate French fries If a child is selected at random, find theprobability that he or she eats French fries

4 In a classroom of 36 students, 8 were liberal arts majors and 7 werehistory majors If a student is selected at random, find the probabilitythat the student is neither a liberal arts nor a history major

5 A recent survey found that 74% of those questioned get some ofthe news from the Internet If a person is selected at random, findthe probability that the person does not get any news from theInternet

CHAPTER 1 Basic Concepts

14

2 The total number of outcomes in this sample space is 50.

a PðAÞ ¼22

50¼

1125

b PðB or ABÞ ¼7 þ 2

50 ¼

950

c P(not O) ¼ 1 19

50¼

3150

d P(neither A nor O) ¼ P(AB or B) ¼2 þ 7

50 ¼

950

3 P(French fries) ¼ 89

356¼

14

4 P(neither liberal arts nor history) ¼ 1 8 þ 7

5 P(does not get any news from the Internet) ¼ 1  0.74 ¼ 0.26

Law of Large Numbers

We know from classical probability that if a coin is tossed one time, we

cannot predict the outcome, but the probability of getting a head is 1

2 andthe probability of getting a tail is12if everything is fair But what happens if

we toss the coin 100 times? Will we get 50 heads? Common sense tells us that

CHAPTER 1 Basic Concepts 15

most of the time, we will not get exactly 50 heads, but we should get close to

50 heads What will happen if we toss a coin 1000 times? Will we getexactly 500 heads? Probably not However, as the number of tosses increases,the ratio of the number of heads to the total number of tosses will getcloser to 1

2 This phenomenon is known as the law of large numbers Thislaw holds for any type of gambling game such as rolling dice, playingroulette, etc

Subjective Probability

A third type of probability is called subjective probability Subjectiveprobability is based upon an educated guess, estimate, opinion, or inexactinformation For example, a sports writer may say that there is a 30%probability that the Pittsburgh Steelers will be in the Super Bowl next year.Here the sports writer is basing his opinion on subjective information such

as the relative strength of the Steelers, their opponents, their coach, etc.Subjective probabilities are used in everyday life; however, they are beyondthe scope of this book

Summary

Probability is the mathematics of chance There are three types

of probability: classical probability, empirical probability, and subjectiveprobability Classical probability uses sample spaces A sample space is theset of outcomes of a probability experiment The range of probability is from

0 to 1 If an event cannot occur, its probability is 0 If an event is certain tooccur, its probability is 1 Classical probability is defined as the number ofways (outcomes) the event can occur divided by the total number ofoutcomes in the sample space

Empirical probability uses frequency distributions, and it is defined as thefrequency of an event divided by the total number of frequencies

Subjective probability is made by a person’s knowledge of the situationand is basically an educated guess as to the chances of an event occurring

CHAPTER 1 Basic Concepts

16

7 The type of probability that uses sample spaces is called

12 If a letter is selected at random from the word ‘‘Mississippi,’’ find the

14 In a survey of 180 people, 74 were over the age of 64 If a person

is selected at random, what is the probability that the person is

15 In a classroom of 24 students, there were 20 freshmen If a student is

selected at random, what is the probability that the student is not a

b 56

c 13

d 16(The answers to the quizzes are found on pages 242–245.)

Probability Sidelight

BRIEF HISTORY OF PROBABILITY

The concepts of probability are as old as humans Paintings in tombsexcavated in Egypt showed that people played games based on chance asearly as 1800 B.C.E One game was called ‘‘Hounds and Jackals’’ and issimilar to the present-day game of ‘‘Snakes and Ladders.’’

Ancient Greeks and Romans made crude dice from various items such

as animal bones, stones, and ivory When some of these items were testedrecently, they were found to be quite accurate These crude dice were alsoused in fortune telling and divination

Emperor Claudius (10 BCE–54 CE) is said to have written a book entitledHow To Win at Dice He liked playing dice so much that he had a special diceboard in his carriage

No formal study of probability was done until the 16th century whenGirolamo Cardano (1501–1576) wrote a book on probability entitled TheBook on Chance and Games Cardano was a philosopher, astrologer,physician, mathematician, and gambler In his book, he also includedtechniques on how to cheat and how to catch others who are cheating He isbelieved to be the first mathematician to formulate a definition of classicalprobability

During the mid-1600s, a professional gambler named Chevalier de Meremade a considerable amount of money on a gambling game He would betunsuspecting patrons that in four rolls of a die, he could obtain at least one 6

He was so successful at winning that word got around, and people refused toplay He decided to invent a new game in order to keep winning Hewould bet patrons that if he rolled two dice 24 times, he would get at leastone double 6 However, to his dismay, he began to lose more often than hewould win and lost money

CHAPTER 1 Basic Concepts

20

Unable to figure out why he was losing, he asked a renowned

mathematician, Blaise Pascal (1623–1662) to study the game Pascal was a

child prodigy when it came to mathematics At the age of 14, he participated

in weekly meetings of the mathematicians of the French Academy At the age

of 16, he invented a mechanical adding machine

Because of the dice problem, Pascal became interested in studying probability

and began a correspondence with a French government official and fellow

mathematician, Pierre de Fermat (1601–1665) Together the two were able to

solve de Mere’s dilemma and formulate the beginnings of probability theory

In 1657, a Dutch mathematician named Christian Huygens wrote a treatise

on the Pascal–Fermat correspondence and introduced the idea of

mathemat-ical expectation (See Chapter 5.)

Abraham de Moivre (1667–1754) wrote a book on probability entitled

Doctrine of Chances in 1718 He published a second edition in 1738

Pierre Simon Laplace (1749–1827) wrote a book and a series of

supplements on probability from 1812 to 1825 His purpose was to acquaint

readers with the theory of probability and its applications, using everyday

language He also stated that the probability that the sun will rise tomorrow

is1,826,214

1,826,215.

Simeon-Denis Poisson (1781–1840) developed the concept of the Poisson

distribution (See Chapter 8.)

Also during the 1800s a mathematician named Carl Friedrich Gauss

(1777–1855) developed the concepts of the normal distribution Earlier work

on the normal distribution was also done by de Moivre and Laplace,

unknown to Gauss (See Chapter 9.)

In 1895, the Fey Manufacturing Company of San Francisco invented the

first automatic slot machine These machines consisted of three wheels that

were spun when a handle on the side of the machine was pulled Each wheel

contained 20 symbols; however, the number of each type of symbols was not

the same on each wheel For example, the first wheel may have 6 oranges,

while the second wheel has 3 oranges, and the third wheel has only one

When a person gets two oranges, the person may think that he has almost

won by getting 2 out of 3 equitable symbols, while the real probability of

winning is much smaller

In the late 1940s, two mathematicians, Jon von Neumann and Stanislaw

Ulam used a computer to simulate probability experiments This method is

called the Monte Carlo method (See Chapter 10.)

Today probability theory is used in insurance, gambling, war gaming, the

stock market, weather forecasting, and many other areas

CHAPTER 1 Basic Concepts 21

point Then from each branch of the first experiment draw branches that

represent the outcomes of the second experiment You can continue the

process for further experiments of the sequence if necessary

EXAMPLE: A coin is tossed and a die is rolled Draw a tree diagram and

find the sample space

SOLUTION:

1 Since there are two outcomes (heads and tails for the coin), draw two

branches from a single point and label one H for head and the other

one T for tail

2 From each one of these outcomes, draw and label six branches

repre-senting the outcomes 1, 2, 3, 4, 5, and 6 for the die

3 Trace through each branch to find the outcomes of the experiment

EXAMPLE: A coin is tossed and a die is rolled Find the probability ofgetting

a A head on the coin and a 3 on the die

b A head on the coin

c Since there are two ways to get a 4 on the die, namely H4 and T4,

Pð4 on the die) ¼ 2

12¼

16

EXAMPLE: Three coins are tossed Draw a tree diagram and find the samplespace

SOLUTION:

Each coin can land either heads up (H) or tails up (T); therefore, the treediagram will consist of three parts and each part will have two branches.See Figure 2-2

Fig 2-2.

CHAPTER 2 Sample Spaces

24

Hence the sample space is HHH, HHT, HTH, HTT, THH, THT, TTH,

TTT

Once the sample space is found, probabilities can be computed

EXAMPLE: Three coins are tossed Find the probability of getting

a Two heads and a tail in any order

b Three heads

c No heads

d At least two tails

e At most two tails

SOLUTION:

a There are eight outcomes in the sample space, and there are three ways

to get two heads and a tail in any order They are HHT, HTH,

d The event of at least two tails means two tails and one head or three

tails There are four outcomes in this event—namely TTH, THT,

HTT, and TTT; hence,

P(at least two tails) ¼4

8¼

12

e The event of getting at most two tails means zero tails, one tail,

or two tails There are seven outcomes in this event—HHH, THH,

HTH, HHT, TTH, THT, and HTT; hence,

P(at most two tails) ¼7

8When selecting more than one object from a group of objects, it is

important to know whether or not the object selected is replaced before

drawing the second object Consider the next two examples

CHAPTER 2 Sample Spaces 25

EXAMPLE: A box contains a red ball (R), a blue ball (B), and a yellow ball(Y) Two balls are selected at random in succession Draw a tree diagram andfind the sample space if the first ball is replaced before the second ball isselected.

SOLUTION:

There are three ways to select the first ball They are a red ball, a blue ball, or

a yellow ball Since the first ball is replaced before the second one is selected,there are three ways to select the second ball They are a red ball, a blue ball,

or a yellow ball The tree diagram is shown in Figure 2-3

The sample space consists of nine outcomes They are RR, RB, RY, BR,

BB, BY, YR, YB, YY Each outcome has a probability of 1

9:Now what happens if the first ball is not replaced before the second ball

is selected?

EXAMPLE: A box contains a red ball (R), a blue ball (B), and a yellow ball(Y) Two balls are selected at random in succession Draw a tree diagram andfind the sample space if the first ball is not replaced before the second ball isselected

SOLUTION:

There are three outcomes for the first ball They are a red ball, a blue ball, or

a yellow ball Since the first ball is not replaced before the second ball isdrawn, there are only two outcomes for the second ball, and these outcomesdepend on the color of the first ball selected If the first ball selected is blue,then the second ball can be either red or yellow, etc The tree diagram isshown in Figure 2-4

Fig 2-3.

CHAPTER 2 Sample Spaces

26

The sample space consists of six outcomes, which are RB, RY, BR, BY,

YR, YB Each outcome has a probability of 1

6:

PRACTICE

1 If the possible blood types are A, B, AB, and O, and each type can be

Rhþ or Rh, draw a tree diagram and find all possible blood types

2 Students are classified as male (M) or female (F), freshman (Fr),

sophomore (So), junior (Jr), or senior (Sr), and full-time (Ft) or

part-time (Pt) Draw a tree diagram and find all possible classifications

3 A box contains a $1 bill, a $5 bill, and a $10 bill Two bills are selected

in succession with replacement Draw a tree diagram and find the

sample space Find the probability that the total amount of money

selected is

a $6

b Greater than $10

c Less than $15

4 Draw a tree diagram and find the sample space for the genders of the

children in a family consisting of 3 children Assume the genders are

equiprobable Find the probability of

a Three girls

b Two boys and a girl in any order

c At least two boys

5 A box contains a white marble (W), a blue marble (B), and a green

marble (G) Two marbles are selected without replacement Draw a

tree diagram and find the sample space Find the probability that one

marble is white

Fig 2-4.

CHAPTER 2 Sample Spaces 27

8 since three girls is GGG.

b P(2 boys and one girl in any order) ¼3

8since there are three ways

to get two boys and one girl in any order They are BBG, BGB,